An Introduction to Mathematical Induction

Stage: 5

(4) The left-hand side is the number of ways of choosing $r$ balls
from $n+1$. Suppose one ball is coloured blue (and the others
aren't). Now explain why the right-hand side is the number of ways
of picking $r$ balls including the blue one plus the number of ways
of picking $r$ balls excluding the blue one.

(5) OK, this isn't true. But the inductive step works. So what's
gone wrong? It's not true for $n=1$. This is why it's absolutely
vital that you check the starting point!

(6) The inductive hypothesis is that $4^k + 6k - 1$ is divisible by
9. That is, $4^k+6k-1=9m$ for some integer $m$. Now use this to get
an expression for $4^k$ that you can substitute into
$4^{k+1}+6(k+1)-1$. Alternatively, what is $4(4^k+6k-1)-18k+9$?
This is more elegant, but perhaps harder to spot without
practice!

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